Confusion about CT scan with contrast

Hey guys!
So since the beginning, I've read that if we are ever experiencing abdominal pain and go to the ER that we have to demand a CT scan with IV contrast. I understand that drinking the liquid will not show a potential obstruction in the bilio-pancreatic limb since food and liquids don't pass through that section. What I don't understand is how exactly an IV contrast will show an obstruction? Doesn't the contrast go through the blood vessels? Will it show the lack of blood flow to part of the biliopancreatic limb if there's an obstruction or is it something else?

And before anyone jumps down my throat, I'm not experiencing any pain, I saw a thread on another site that was upsetting and just got me thinking that I don't 100% understand how it works.
Thanks!

But this is the WHAT TO DO part. Print it out (large), carry a copy. Drag a copy to a thumb drive. Not many places will let you use your thumb drive in their computer, but there is a chance. If you drag an iPad with you everywhere like I do, a copy there wouldn't hurt.

How does a CT scan work?
A CT scan works similar to an x-ray. The body casts a "shadow" on film when it is exposed to the x-ray, much like when you hold a flashlight up to your hand and cast a shadow on a wall. All of the tissue that the x-ray passes through overlap on the image, making it hard to isolate different elements. A CT scan works around this limitation by capturing one narrow slice of your body at a time. Inside the CT machine, the x-ray tube circles around the patient taking pictures as it rotates. These slices can be viewed two-dimensionally or added back together to create a three-dimensional image of a body structure.

A dye (contrast agent) may be injected into your bloodstream to enhance certain body tissues. The dye contains iodine, a substance that x-rays cannot pass through. It circulates through the blood stream and is absorbed in certain tissues, which then stand out on the scan.

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